Thursday, March 25, 2010

A little while back I got into a discussion of Wittgenstein at PhilPapers. The issue was mainly about whether or not the instigator of the discussion, Professor Jim Stone, had successfully undermined Wittgenstein's notion of family resemblance in a paper he had published in the '90s. (Professor Stone started the discussion thread because he felt his paper had gone unnoticed in the annals of philosophical research, and that it deserved more attention.) Though the notion of family resemblance is more or less well-established in some circles, Professor Stone (wrongly) claims that it is supported solely by the discussion of games and numbers found in Wittgenstein's Philosophical Investigations. (The most immediately relevant sections of the Investigations can be found online, with commentary by Lois Shawver, here.--I have not read Shawver's commentary, however, so do not assume I agree with it.) Stone says that W.'s point about numbers is not commonly respected, and so he does not bother addressing it. The only issue is games, and he attempts to resolve that issue by giving a definition of the concept of "game," thereby disproving W.'s thesis that no such definition may be given.

Much of the discussion is about whether or not Professor Stone gives an adequate account of games. I do not think he does. However, that is not the question I want to address here. The more interesting question, I think, is how to properly understand Wittgenstein. Professor Stone misunderstands Wittgenstein by supposing that to disprove Wittgenstein one need only give a definition of game. I explained this at some length in the discussion thread, and have posted some of my comments below. Again, there's some interesting discussion about how to define "game," so I recommend checking out the whole thread. What I am posting below is only meant to consolidate some of my views on how to understand Wittgenstein and his notion of family resemblance.

Oct. 24, 2009:Wittgenstein's point was not that we cannot draw a clear line around the concept of "game," nor was it that such lines could not be interesting or valuable. It was rather that such lines do not precede our use of the expression "game." In order to circumscribe the concept of "game," we must draw distinctions which have not already been made, and which do not underlie every meaningful and valid use of the concept. We need not first have a clearly defined concept before we can meaningfully use an expression. The uses of ordinary language expressions have developed in the absence of clear definitions which fully account for their use.

Oct. 30, 2009:You refer to W.'s comment "don't think, but look!" as an injunction, when it was no such thing--as if W. universally condemned thought about games. The Investigations are more commonly interpreted as a series of tasks which the reader is asked to undertake in order to recover from their (presumed) restricted view of language and meaning. When W. asks us not to think, but to look, he is asking us to perform a specific task for a specific purpose. By asking us to look at what we call games and consider what we see before defining a strict concept of a game, he asks us to consider all the sorts of things we call "games" without first deciding what defines a game as such. He wants us to think about games, but not by analyzing them in terms of a single rule. The point is that we can (and normally do) think about games without analyzing them in terms of an all-encompassing rule, and that if we do draw such a rule, it will (at best) only resemble the way we had previously thought of games, much in the way that a vaguely defined set of color patches only resembles a set of patches with clear lines of demarcation (PI, paragraph 76).

W. acknowledges (for example, in PI, paragraphs 68 and 69) that we can draw boundaries and so define "game" for any number of purposes. So, again, while your rule for "games" is interesting and perhaps valuable, it does not challenge W.'s point, which is that the use of our concepts does not depend on the prior drawing of all-inclusive rules. As he says (PI, paragraph 68): "[The use of the word 'game'] is not everywhere circumscribed by rules; but no more are there any rules for how high one throws the ball in tennis, or how hard; yet tennis is a game for all that and has rules too." We have rules for the word "game," but they do not cover every aspect of our use of that word, just as the rules of tennis do not cover every aspect of our behavior during that game.

So, again, if your position is that there are various, loose, and overlapping uses of the term "game," and that your definition circumscribes one such use, then you have not challenged Wittgenstein.

. . .

You write, summarizing what you take to be W.'s insight into the nature of language: "If language is made of games, however, what counts as success in playing a language game has no independent or external value or validity; succeeding is wholly internal to the game, which is what defines success."

Firstly, I do not think W. ever says that languages are made of games. He notes similarities between languages and games, and adopts the term "language-game" to reflect the lack of specifity here. I do not think he means to refer to all languages as games, or as collections of games. More importantly, W. does not claim that all languages (or all games, as I noted earlier) are completely circumscribed by rules. Quite the contrary. The value of speaking a language is not limited to or wholly determined by what is deemed correct by the rules of that language. So I do not think you have correctly summarized W.'s insight.

Nov. 19, 2009:I suppose it is commonly said that Wittgenstein's notion of family resemblance rests on the examples of games and numbers, but one can find many other examples of its application in his Philosophical Investigations: for example, with respect to knowing/understanding (paragraphs 143-155), reading (paragraphs 156 - 171), and being guided (172 - 180). Wittgenstein may not offer these arguments to substantiate his notion of family resemblance, but they are clearly applications of that notion, and so any criticism of that notion should also take such applications into account. Merely criticizing his discussion of games (or numbers) is not enough.

Dec. 1, 2009:Wittgenstein quite clearly says we do not need a definition which accounts for every legitimate use of the term "game." Consider these excerpts:

PI, 68: "For how is the concept of a game bounded? What still counts as a game and what no longer does? Can you give the boundary? No. You can draw one; for none has so far been drawn. (But that never troubled you before when you used the word 'game.')"

PI, 69: "How should we explain to someone what a game is? I imagine that we should describe games to him, and we might add: 'This and similar things are called 'games'. And do we know any more about it ourselves? Is it only other people whom we cannot tell exactly what a game is?--But this is not ignorance. We do not know the boundaries because none have been drawn. To repeat, we can draw a boundary--for a special purpose. Does it take that to make the concept usable? Not at all! (Except for the special purpose.)"

Wittgenstein says that one can draw a definition for the concept of game, but that such a definition was not needed for the concept to be useful.

He makes the same point with the concept of "plants": "When I give the description: 'The ground was covered with plants'--do you want to say I don't know what I am talking about until I can give a definition of a plant?" (PI, 70)

Now, it may very well be that all games do have something in common--but having such a feature is not what makes them games. It would be a remarkable coincidence if everything we called a game had one feature in common which was not shared by anything else. For we do not appeal to such a feature when we apply or judge applications of the term "game." That is, unless we are using a strict definition for a special purpose, as W. notes.

So, an interpretation that says "Well, Ludwig didn't mean to deny that games share a common feature in virtue of which they are games, only that that isn't how our concept works; so if you come up with the feature, it wasn't needed"--such an interpretation is surely false. Wittgenstein does deny that all games share a common feature by virtue of which they are games. But this does not mean that games cannot share a common feature. So, pointing out a common feature does not in any way challenge Wittgenstein's point.

Dec. 4, 2009:W. takes the notion of family resemblance to indicate sets which are not bound together by a common element, but which are grouped according to various overlapping similarities. Our notions of such sets are family resemblance concepts, which are employed in the absence of definitions which fully account for their use. Yet, this does not discount the possibility of drawing definitions to suit our needs. As far as interpreting W. goes, we seem to agree on this much, I think.

It is important to dwell for a moment on what constitutes the grouping of a family resemblance set. The grouping is something we do. It does not precede our use of the family resemblance concept. Rather, it is constituted by our use of that concept. Family resemblance sets are grouped by application of the concepts which indicate them. The concepts determine the sets. Thus, Wittgenstein regards games as a family resemblance set because the concept of games is a family resemblance concept.

If you were to turn it around and claim that "game" is a family resemblance concept because the set of all games lacks a common, defining ingredient, it would mean the set of all games existed independently of our categorization, as if there were a definable set of all games and we just didn't know its boundaries. Yet, as Wittgenstein says, our inability to give a definition for "game" is not a result of our ignorance. It is a result of the fact that no definition has been given. (See the quotes I offered in my last post.) We cannot isolate the set of all games because no well-defined set exists as such. But, as W. says, we can define such a set for a special purpose. The application of the concept "game" is not realized in the world ahead of time. We can thus disagree about how to apply the concept of "games" without being able to appeal to a rule to decide who is right--again, not out of ignorance, but because the concept is not completely circumscribed by rules.

You seem to take it as trivial that W. claims we can employ concepts in the absence of all-encompassing definitions. Yet, if you will grant this trivial point, Wittgenstein's conclusion seems to follow. That is, unless you suppose that our employment of concepts mysteriously matches with independent rules for their application, even if we do not know of them; in which case, if we do disagree on the application of a concept, there is a sense in which one of us is right, and the other is wrong, even though we have no means of knowing it. Wittgenstein clearly does not embrace such a view, and neither do I. There is no basis for regarding the set of all games apart from the rules we use to regard that set. If our rules are not all-encompassing, then the set of games is a family resemblance set.

Your objection is that there really is a common feature for all games, and that we have such a feature in mind (somehow, perhaps unconsciously) when we talk about games. However, this is highly unlikely, considering the way we learn how to use the term "games." Even if there is a common feature, such a feature is not obviously what defines games as such. It seems more likely that such a feature would be wholly coincidental, since we do not appeal to it and are so far unaware of it, despite our complex use of the term "game." (Consider why it is that W. constantly reminds us to look at how our language is used and learned.) Thus, W. predicts that no such feature is to be found. But to overcome W., it is not enough to suggest a common feature shared by all games. You would rather have to find a feature which defined games as such. That is, you would have to show that the feature in question was operant in our conceptualization of games. To undermine the notion of family resemblance (as it applies to games), you would have to show that our grouping together of games relies on some recognition of that shared feature.

Of course, even if you were to succeed in demonstrating that "game" is not a family resemblance concept (and I wish you luck on that quest, though I do not expect you will succeed), you will not have undermined the notion of family resemblance. For the notion has many other applications, both in and apart from Wittgenstein's Investigations.

Dec. 6, 2009:As you say, to challenge Wittgenstein's family-resemblance analysis of games, you must show that a single definition picks out something common to all games and only games, and that it "seems intuitive that this is what makes them games." It should be just obvious that the proposed definition is what we mean by the word "game." Yet, it is only obvious that we might mean something sort of like what you propose in some cases, and not all, and that there is no established boundary between the use you propose and other uses. So I do not see how your definition, if cogent, would require any change to what Wittgenstein has said. Furthermore, several problems with the application of your proposed definition have been noted, and I do not think you have addressed all of them. So your definition is not particularly compelling or intuitively appealing, regardless of whether or not it challenges W.

Also, I do not think Wittgenstein would disagree with your claim that "we are sometimes guided by commonalities that we do not know how to express." Indeed, we are guided by commonalities. That is the point. We are so guided, but we do not have rules ahead of time which define their boundaries. We do not always know how and when to apply them appropriately--and this is not a matter of ignorance, but simply a lack of definition. What lends credence to Wittgenstein's view is not merely the fact that we do not learn language solely by applying dictionary definitions, though that is certainly evidence in W.'s favor. Rather, W. points out problems which arise when you try to think how the conception of "game" could somehow fully account for its use ahead of time, as if all of the possible movements of a machine were there in the machine before it was ever used. Once we realize that the use of a term is not determined in advance, a whole host of philosophical problems disappear.

To add more weight to your rejection of W., you mention rectangles as a case of a non-family-resemblance concept. Here's how I think W. would respond: Imagine a series of pictures of a thickly drawn red rectangle against an orange background. In each successive picture, the rectangle becomes more and more blurred (or, to take it another way, becomes broken apart into smaller and smaller fragments). At some point in this series, it is obvious that there is no longer a picture of a rectangle. But at what point can you say "this is the first picture at which the rectangle is present [or absent]?" We have no rule for that. Whether or not something is a rectangle is clear enough in many cases, so that the use of the concept is firmly established; but we don't have rules for all cases.

Now imagine the case of a rectangle drawn on a sphere. In some sense, we might say it is impossible to have a rectangle on a sphere; yet, in many cases, we can say that it is a rectangle without any trouble. Indeed, if you draw a rectangle on the ground, have you not drawn it on something much like a sphere? At what point do we say the spherical diagram is not a rectangle? Again, we have no rule for that.

Dec. 8, 2009:I think that, to successfully counter W., your definition must obviously be the one we have always had in mind--once it is understood, of course. That is, once we grasp the correct application of your definition, we should not hesitate to accept that this is the way we have always used the word "game." We should not require any additional evidence or argument to convince us. Thus, as I said, it should just be obvious. That is how I understand the situation, and I thought you did as well. That is why I interpreted your use of the expression "seem intuitive" as I did. If you meant "seem intuitive" to mean something else, I hope you don't mind explaining how it differs from what I have said.

It is interesting where you agree and disagree with W. on vagueness. For, wouldn't you agree that the vagueness inherent in some of our concepts would naturally lead us towards divergent applications of them? So that our application of those concepts does not itself constitute a well-defined set, but rather a set of entities which contain many overlapping commonalities?

As Wittgenstein says (forgive me for not looking up the exact quote right now), when we see a blurry picture of somebody from far off, we do not correctly describe it by saying what they look like up close. The point, I think, is that a well-defined rule does not accurately describe what our vague concepts look like.

You note that we are adept at triangulation, and that we can be clever in the way we are guided by commonalities. But, if the concept is vague, as you seem to allow, then no clear definition will exactly indicate the concept we had previously been using.

Thus, your departure from W. is hard to understand. It occurs here, when you say, " One test of the adequacy of the definition is that it will explain our ambivalence in such cases." I don't see how that works. Can a vague definition account for its own vagueness?

A little while back I got into a discussion of Wittgenstein at PhilPapers. The issue was mainly about whether or not the instigator of the discussion, Professor Jim Stone, had successfully undermined Wittgenstein's notion of family resemblance in a paper he had published in the '90s. (Professor Stone started the discussion thread because he felt his paper had gone unnoticed in the annals of philosophical research, and that it deserved more attention.) Though the notion of family resemblance is more or less well-established in some circles, Professor Stone (wrongly) claims that it is supported solely by the discussion of games and numbers found in Wittgenstein's Philosophical Investigations. (The most immediately relevant sections of the Investigations can be found online, with commentary by Lois Shawver, here.--I have not read Shawver's commentary, however, so do not assume I agree with it.) Stone says that W.'s point about numbers is not commonly respected, and so he does not bother addressing it. The only issue is games, and he attempts to resolve that issue by giving a definition of the concept of "game," thereby disproving W.'s thesis that no such definition may be given.

Much of the discussion is about whether or not Professor Stone gives an adequate account of games. I do not think he does. However, that is not the question I want to address here. The more interesting question, I think, is how to properly understand Wittgenstein. Professor Stone misunderstands Wittgenstein by supposing that to disprove Wittgenstein one need only give a definition of game. I explained this at some length in the discussion thread, and have posted some of my comments below. Again, there's some interesting discussion about how to define "game," so I recommend checking out the whole thread. What I am posting below is only meant to consolidate some of my views on how to understand Wittgenstein and his notion of family resemblance.

Oct. 24, 2009:Wittgenstein's point was not that we cannot draw a clear line around the concept of "game," nor was it that such lines could not be interesting or valuable. It was rather that such lines do not precede our use of the expression "game." In order to circumscribe the concept of "game," we must draw distinctions which have not already been made, and which do not underlie every meaningful and valid use of the concept. We need not first have a clearly defined concept before we can meaningfully use an expression. The uses of ordinary language expressions have developed in the absence of clear definitions which fully account for their use.

Oct. 30, 2009:You refer to W.'s comment "don't think, but look!" as an injunction, when it was no such thing--as if W. universally condemned thought about games. The Investigations are more commonly interpreted as a series of tasks which the reader is asked to undertake in order to recover from their (presumed) restricted view of language and meaning. When W. asks us not to think, but to look, he is asking us to perform a specific task for a specific purpose. By asking us to look at what we call games and consider what we see before defining a strict concept of a game, he asks us to consider all the sorts of things we call "games" without first deciding what defines a game as such. He wants us to think about games, but not by analyzing them in terms of a single rule. The point is that we can (and normally do) think about games without analyzing them in terms of an all-encompassing rule, and that if we do draw such a rule, it will (at best) only resemble the way we had previously thought of games, much in the way that a vaguely defined set of color patches only resembles a set of patches with clear lines of demarcation (PI, paragraph 76).

W. acknowledges (for example, in PI, paragraphs 68 and 69) that we can draw boundaries and so define "game" for any number of purposes. So, again, while your rule for "games" is interesting and perhaps valuable, it does not challenge W.'s point, which is that the use of our concepts does not depend on the prior drawing of all-inclusive rules. As he says (PI, paragraph 68): "[The use of the word 'game'] is not everywhere circumscribed by rules; but no more are there any rules for how high one throws the ball in tennis, or how hard; yet tennis is a game for all that and has rules too." We have rules for the word "game," but they do not cover every aspect of our use of that word, just as the rules of tennis do not cover every aspect of our behavior during that game.

So, again, if your position is that there are various, loose, and overlapping uses of the term "game," and that your definition circumscribes one such use, then you have not challenged Wittgenstein.

. . .

You write, summarizing what you take to be W.'s insight into the nature of language: "If language is made of games, however, what counts as success in playing a language game has no independent or external value or validity; succeeding is wholly internal to the game, which is what defines success."

Firstly, I do not think W. ever says that languages are made of games. He notes similarities between languages and games, and adopts the term "language-game" to reflect the lack of specifity here. I do not think he means to refer to all languages as games, or as collections of games. More importantly, W. does not claim that all languages (or all games, as I noted earlier) are completely circumscribed by rules. Quite the contrary. The value of speaking a language is not limited to or wholly determined by what is deemed correct by the rules of that language. So I do not think you have correctly summarized W.'s insight.

Nov. 19, 2009:I suppose it is commonly said that Wittgenstein's notion of family resemblance rests on the examples of games and numbers, but one can find many other examples of its application in his Philosophical Investigations: for example, with respect to knowing/understanding (paragraphs 143-155), reading (paragraphs 156 - 171), and being guided (172 - 180). Wittgenstein may not offer these arguments to substantiate his notion of family resemblance, but they are clearly applications of that notion, and so any criticism of that notion should also take such applications into account. Merely criticizing his discussion of games (or numbers) is not enough.

Dec. 1, 2009:Wittgenstein quite clearly says we do not need a definition which accounts for every legitimate use of the term "game." Consider these excerpts:

PI, 68: "For how is the concept of a game bounded? What still counts as a game and what no longer does? Can you give the boundary? No. You can draw one; for none has so far been drawn. (But that never troubled you before when you used the word 'game.')"

PI, 69: "How should we explain to someone what a game is? I imagine that we should describe games to him, and we might add: 'This and similar things are called 'games'. And do we know any more about it ourselves? Is it only other people whom we cannot tell exactly what a game is?--But this is not ignorance. We do not know the boundaries because none have been drawn. To repeat, we can draw a boundary--for a special purpose. Does it take that to make the concept usable? Not at all! (Except for the special purpose.)"

Wittgenstein says that one can draw a definition for the concept of game, but that such a definition was not needed for the concept to be useful.

He makes the same point with the concept of "plants": "When I give the description: 'The ground was covered with plants'--do you want to say I don't know what I am talking about until I can give a definition of a plant?" (PI, 70)

Now, it may very well be that all games do have something in common--but having such a feature is not what makes them games. It would be a remarkable coincidence if everything we called a game had one feature in common which was not shared by anything else. For we do not appeal to such a feature when we apply or judge applications of the term "game." That is, unless we are using a strict definition for a special purpose, as W. notes.

So, an interpretation that says "Well, Ludwig didn't mean to deny that games share a common feature in virtue of which they are games, only that that isn't how our concept works; so if you come up with the feature, it wasn't needed"--such an interpretation is surely false. Wittgenstein does deny that all games share a common feature by virtue of which they are games. But this does not mean that games cannot share a common feature. So, pointing out a common feature does not in any way challenge Wittgenstein's point.

Dec. 4, 2009:W. takes the notion of family resemblance to indicate sets which are not bound together by a common element, but which are grouped according to various overlapping similarities. Our notions of such sets are family resemblance concepts, which are employed in the absence of definitions which fully account for their use. Yet, this does not discount the possibility of drawing definitions to suit our needs. As far as interpreting W. goes, we seem to agree on this much, I think.

It is important to dwell for a moment on what constitutes the grouping of a family resemblance set. The grouping is something we do. It does not precede our use of the family resemblance concept. Rather, it is constituted by our use of that concept. Family resemblance sets are grouped by application of the concepts which indicate them. The concepts determine the sets. Thus, Wittgenstein regards games as a family resemblance set because the concept of games is a family resemblance concept.

If you were to turn it around and claim that "game" is a family resemblance concept because the set of all games lacks a common, defining ingredient, it would mean the set of all games existed independently of our categorization, as if there were a definable set of all games and we just didn't know its boundaries. Yet, as Wittgenstein says, our inability to give a definition for "game" is not a result of our ignorance. It is a result of the fact that no definition has been given. (See the quotes I offered in my last post.) We cannot isolate the set of all games because no well-defined set exists as such. But, as W. says, we can define such a set for a special purpose. The application of the concept "game" is not realized in the world ahead of time. We can thus disagree about how to apply the concept of "games" without being able to appeal to a rule to decide who is right--again, not out of ignorance, but because the concept is not completely circumscribed by rules.

You seem to take it as trivial that W. claims we can employ concepts in the absence of all-encompassing definitions. Yet, if you will grant this trivial point, Wittgenstein's conclusion seems to follow. That is, unless you suppose that our employment of concepts mysteriously matches with independent rules for their application, even if we do not know of them; in which case, if we do disagree on the application of a concept, there is a sense in which one of us is right, and the other is wrong, even though we have no means of knowing it. Wittgenstein clearly does not embrace such a view, and neither do I. There is no basis for regarding the set of all games apart from the rules we use to regard that set. If our rules are not all-encompassing, then the set of games is a family resemblance set.

Your objection is that there really is a common feature for all games, and that we have such a feature in mind (somehow, perhaps unconsciously) when we talk about games. However, this is highly unlikely, considering the way we learn how to use the term "games." Even if there is a common feature, such a feature is not obviously what defines games as such. It seems more likely that such a feature would be wholly coincidental, since we do not appeal to it and are so far unaware of it, despite our complex use of the term "game." (Consider why it is that W. constantly reminds us to look at how our language is used and learned.) Thus, W. predicts that no such feature is to be found. But to overcome W., it is not enough to suggest a common feature shared by all games. You would rather have to find a feature which defined games as such. That is, you would have to show that the feature in question was operant in our conceptualization of games. To undermine the notion of family resemblance (as it applies to games), you would have to show that our grouping together of games relies on some recognition of that shared feature.

Of course, even if you were to succeed in demonstrating that "game" is not a family resemblance concept (and I wish you luck on that quest, though I do not expect you will succeed), you will not have undermined the notion of family resemblance. For the notion has many other applications, both in and apart from Wittgenstein's Investigations.

Dec. 6, 2009:As you say, to challenge Wittgenstein's family-resemblance analysis of games, you must show that a single definition picks out something common to all games and only games, and that it "seems intuitive that this is what makes them games." It should be just obvious that the proposed definition is what we mean by the word "game." Yet, it is only obvious that we might mean something sort of like what you propose in some cases, and not all, and that there is no established boundary between the use you propose and other uses. So I do not see how your definition, if cogent, would require any change to what Wittgenstein has said. Furthermore, several problems with the application of your proposed definition have been noted, and I do not think you have addressed all of them. So your definition is not particularly compelling or intuitively appealing, regardless of whether or not it challenges W.

Also, I do not think Wittgenstein would disagree with your claim that "we are sometimes guided by commonalities that we do not know how to express." Indeed, we are guided by commonalities. That is the point. We are so guided, but we do not have rules ahead of time which define their boundaries. We do not always know how and when to apply them appropriately--and this is not a matter of ignorance, but simply a lack of definition. What lends credence to Wittgenstein's view is not merely the fact that we do not learn language solely by applying dictionary definitions, though that is certainly evidence in W.'s favor. Rather, W. points out problems which arise when you try to think how the conception of "game" could somehow fully account for its use ahead of time, as if all of the possible movements of a machine were there in the machine before it was ever used. Once we realize that the use of a term is not determined in advance, a whole host of philosophical problems disappear.

To add more weight to your rejection of W., you mention rectangles as a case of a non-family-resemblance concept. Here's how I think W. would respond: Imagine a series of pictures of a thickly drawn red rectangle against an orange background. In each successive picture, the rectangle becomes more and more blurred (or, to take it another way, becomes broken apart into smaller and smaller fragments). At some point in this series, it is obvious that there is no longer a picture of a rectangle. But at what point can you say "this is the first picture at which the rectangle is present [or absent]?" We have no rule for that. Whether or not something is a rectangle is clear enough in many cases, so that the use of the concept is firmly established; but we don't have rules for all cases.

Now imagine the case of a rectangle drawn on a sphere. In some sense, we might say it is impossible to have a rectangle on a sphere; yet, in many cases, we can say that it is a rectangle without any trouble. Indeed, if you draw a rectangle on the ground, have you not drawn it on something much like a sphere? At what point do we say the spherical diagram is not a rectangle? Again, we have no rule for that.

Dec. 8, 2009:I think that, to successfully counter W., your definition must obviously be the one we have always had in mind--once it is understood, of course. That is, once we grasp the correct application of your definition, we should not hesitate to accept that this is the way we have always used the word "game." We should not require any additional evidence or argument to convince us. Thus, as I said, it should just be obvious. That is how I understand the situation, and I thought you did as well. That is why I interpreted your use of the expression "seem intuitive" as I did. If you meant "seem intuitive" to mean something else, I hope you don't mind explaining how it differs from what I have said.

It is interesting where you agree and disagree with W. on vagueness. For, wouldn't you agree that the vagueness inherent in some of our concepts would naturally lead us towards divergent applications of them? So that our application of those concepts does not itself constitute a well-defined set, but rather a set of entities which contain many overlapping commonalities?

As Wittgenstein says (forgive me for not looking up the exact quote right now), when we see a blurry picture of somebody from far off, we do not correctly describe it by saying what they look like up close. The point, I think, is that a well-defined rule does not accurately describe what our vague concepts look like.

You note that we are adept at triangulation, and that we can be clever in the way we are guided by commonalities. But, if the concept is vague, as you seem to allow, then no clear definition will exactly indicate the concept we had previously been using.

Thus, your departure from W. is hard to understand. It occurs here, when you say, " One test of the adequacy of the definition is that it will explain our ambivalence in such cases." I don't see how that works. Can a vague definition account for its own vagueness?